Initial notes on open/closed addressing.

2024-06-12 07:35:13 -06:00 · 2024-06-12 07:35:13 -06:00 · 9045439d0b
parent 7664346b62
commit 9045439d0b
15 changed files with 415 additions and 50 deletions
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--- a/notes/_journal/2024-06-11.md
+++ b/notes/_journal/2024-06-11.md
@ -1,9 +0,0 @@
---
-title: "2024-06-11"
---
-
- [x] Anki Flashcards
- [x] KoL
- [x] OGS
- [ ] Sheet Music (10 min.)
- [ ] Korean (Read 1 Story)
--- a/notes/_journal/2024-06-12.md
+++ b/notes/_journal/2024-06-12.md
@ -0,0 +1,11 @@
+---
+title: "2024-06-12"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* Notes on [[open-addressing|open]] and [[closed-addressing|closed]] hashing.
--- a/notes/_journal/2024-06/2024-06-11.md
+++ b/notes/_journal/2024-06/2024-06-11.md
@ -0,0 +1,12 @@
+---
+title: "2024-06-11"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [x] OGS
+- [ ] Sheet Music (10 min.)
+- [ ] Korean (Read 1 Story)
+
+* Cartesian product [[set#Cancellation Laws|cancellation laws]].
+* Notes on [[relations]].
--- a/notes/algebra/set.md
+++ b/notes/algebra/set.md
@ -27,6 +27,85 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1716396060605-->
 END%%

+## Symmetric Difference
+
+Define the **symmetric difference** of sets $A$ and $B$ as $$A \mathop{\triangle} B = (A - B)  \cup (B - A)$$
+
+%%ANKI
+Basic
+What two operators are used in the definition of the symmetric difference?
+Back: $\cup$ and $-$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1717554445662-->
+END%%
+
+%%ANKI
+Basic
+How is the symmetric difference of sets $A$ and $B$ denoted?
+Back: $A \mathop{\triangle} B$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1717554445665-->
+END%%
+
+%%ANKI
+Basic
+How is $A \mathop{\triangle} B$ defined?
+Back: As $(A - B)  \cup (B - A)$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1717554445670-->
+END%%
+
+## Cartesian Product
+
+Given two sets $A$ and $B$, the **Cartesian product** $A \times B$ is defined as: $$A \times B = \{\langle x, y \rangle \mid x \in A \land y \in B\}$$
+
+%%ANKI
+Basic
+How is the Cartesian product of $A$ and $B$ denoted?
+Back: $A \times B$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1717679397781-->
+END%%
+
+%%ANKI
+Basic
+Using ordered pairs, how is $A \times B$ defined?
+Back: $\{\langle x, y \rangle \mid x \in A \land y \in B\}$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1717679397797-->
+END%%
+
+%%ANKI
+Basic
+Who is attributed the representation of points in a plane?
+Back: René Descartes.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1717679397825-->
+END%%
+
+%%ANKI
+Basic
+Why is the Cartesian product named the way it is?
+Back: It is named after René Descartes.
+Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
+<!--ID: 1717679397836-->
+END%%
+
+%%ANKI
+Basic
+Suppose $x, y \in A$. What set is $\langle x, y \rangle$ in?
+Back: $\mathscr{P}\mathscr{P}A$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1717679397848-->
+END%%
+
+%%ANKI
+Cloze
+{$x \in A$} iff {$\{x\} \subseteq A$} iff {$\{x\} \in \mathscr{P}A$}.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1717679397860-->
+END%%
+
 ## Laws

 The algebra of sets obey laws reminiscent (but not exactly) of the algebra of real numbers.
@ -676,32 +755,43 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1717073537007-->
 END%%

-## Symmetric Difference
+### Cancellation Laws

-Define the **symmetric difference** of sets $A$ and $B$ as $$A \mathop{\triangle} B = (A - B)  \cup (B - A)$$
+Let $A$, $B$, and $C$ be sets. If $A \neq \varnothing$,
+
+* $(A \times B = A \times C) \Rightarrow B = C$
+* $(B \times A = C \times A) \Rightarrow B = C$

 %%ANKI
 Basic
-What two operators are used in the definition of the symmetric difference?
-Back: $\cup$ and $-$.
+What is the left cancellation law of the Cartesian product?
+Back: $(A \times B = A \times C) \Rightarrow B = C$
 Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
-<!--ID: 1717554445662-->
+<!--ID: 1718107987907-->
 END%%

 %%ANKI
 Basic
-How is the symmetric difference of sets $A$ and $B$ denoted?
-Back: $A \mathop{\triangle} B$
+$(A \times B = A \times C) \Rightarrow B = C$ is always true if what condition is satisfied?
+Back: $A \neq \varnothing$
 Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
-<!--ID: 1717554445665-->
+<!--ID: 1718107987918-->
 END%%

 %%ANKI
 Basic
-How is $A \mathop{\triangle} B$ defined?
-Back: As $(A - B)  \cup (B - A)$.
+What is the right cancellation law of the Cartesian product?
+Back: $(B \times A = C \times A) \Rightarrow B = C$
 Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
-<!--ID: 1717554445670-->
+<!--ID: 1718107987928-->
+END%%
+
+%%ANKI
+Basic
+$(B \times A = C \times A) \Rightarrow B = C$ is always true if what condition is satisfied?
+Back: $A \neq \varnothing$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718107987936-->
 END%%

 ## Bibliography
--- a/notes/hashing/closed-addressing.md
+++ b/notes/hashing/closed-addressing.md
@ -0,0 +1,55 @@
+---
+title: Closed Addressing
+TARGET DECK: Obsidian::STEM
+FILE TAGS: hashing::closed
+tags:
+  - hashing
+---
+
+## Overview
+
+In **closed addressing**, a key is always stored in the bucket it's hashed to. Collisions are dealt with using separate data structures on a per-bucket basis.
+
+%%ANKI
+Basic
+What does "closed" refer to in term "closed addressing"?
+Back: A key is always stored in the slot it hashes to.
+Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
+<!--ID: 1718198717474-->
+END%%
+
+%%ANKI
+Basic
+What does "open" refer to in term "open hashing"?
+Back: A key may resides in a data structure separate from the hash table.
+Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
+<!--ID: 1718198717484-->
+END%%
+
+%%ANKI
+Cloze
+{Closed} addressing is also known as {open} hashing.
+Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
+<!--ID: 1718198717495-->
+END%%
+
+%%ANKI
+Cloze
+The following is an example of {closed} addressing.
+![[closed-addressing.png]]
+Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
+<!--ID: 1718198717506-->
+END%%
+
+%%ANKI
+Cloze
+The following is an example of {open} hashing.
+![[closed-addressing.png]]
+Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
+<!--ID: 1718198755496-->
+END%%
+
+## Bibliography
+
+* “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
+* Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
--- a/notes/hashing/direct-addressing.md
+++ b/notes/hashing/direct-addressing.md
@ -121,6 +121,23 @@ Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (
 <!--ID: 1716307180987-->
 END%%

+%%ANKI
+Basic
+What distinguishes direct addressing from closed and open addressing?
+Back: Direct addressing isn't concerned with conflicting keys.
+Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
+<!--ID: 1718199205862-->
+END%%
+
+%%ANKI
+Basic
+Direct addressing sits between what other addressing types?
+Back: Open and closed addressing.
+Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
+<!--ID: 1718199205872-->
+END%%
+
 ## Bibliography

+* “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
 * Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
--- a/notes/hashing/images/closed-addressing.png
+++ b/notes/hashing/images/closed-addressing.png
--- a/notes/hashing/images/open-addressing.png
+++ b/notes/hashing/images/open-addressing.png
--- a/notes/hashing/index.md
+++ b/notes/hashing/index.md
@ -137,6 +137,48 @@ Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (
 <!--ID: 1716307180980-->
 END%%

+An **independent uniform hash function** is the ideal theoretical abstraction. For each possible input $k$ in universe $U$, an output $h(k)$ is produced randomly and independently chosen from range $\{0, 1, \ldots, m - 1\}$. Once a value $h(k)$ is chosen, each subsequent call to $h$ with the same input $k$ yields the same output $h(k)$.
+
+%%ANKI
+Basic
+What is considered the ideal (though only theoretical) hash function?
+Back: The independent uniform hash function.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1718197741507-->
+END%%
+
+%%ANKI
+Basic
+Given independent uniform hash function $h$, what about $h$ is "independent"?
+Back: Each key $k$ has output $h(k)$ determined independently from other keys.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1718197741527-->
+END%%
+
+%%ANKI
+Basic
+Given independent uniform hash function $h$, what about $h$ is "uniform"?
+Back: Every output of $h$ is equally likely to be any of the values in its range.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1718197741537-->
+END%%
+
+%%ANKI
+Basic
+With respect to hashing, a random oracle refers to what kind of hash function?
+Back: An independent uniform hash function.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1718197741545-->
+END%%
+
+%%ANKI
+Basic
+Where does "memory" come into play with independent uniform hash functions?
+Back: Once $h(k)$ is determined, subsequent calls to $h$ with $k$ always yield the same value.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1718197741555-->
+END%%
+
 ## Bibliography

 * Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
--- a/notes/hashing/open-addressing.md
+++ b/notes/hashing/open-addressing.md
@ -0,0 +1,55 @@
+---
+title: Open Addressing
+TARGET DECK: Obsidian::STEM
+FILE TAGS: hashing::open
+tags:
+  - hashing
+---
+
+## Overview
+
+In **open addressing**, keys always reside in the hash table. Collisions are dealt with by searching for other empty buckets within the hash table.
+
+%%ANKI
+Basic
+What does "closed" refer to in term "closed hashing"?
+Back: A key must reside in the hash table.
+Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
+<!--ID: 1718198717434-->
+END%%
+
+%%ANKI
+Basic
+What does "open" refer to in term "open addressing"?
+Back: A key is not necessarily stored in the slot it hashes to.
+Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
+<!--ID: 1718198717447-->
+END%%
+
+%%ANKI
+Cloze
+{Open} addressing is also known as {closed} hashing.
+Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
+<!--ID: 1718198717455-->
+END%%
+
+%%ANKI
+Cloze
+The following is an example of {closed} hashing.
+![[open-addressing.png]]
+Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
+<!--ID: 1718198717464-->
+END%%
+
+%%ANKI
+Cloze
+The following is an example of {open} addressing.
+![[open-addressing.png]]
+Reference: “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
+<!--ID: 1718198755486-->
+END%%
+
+## Bibliography
+
+* “Hash Tables: Open vs Closed Addressing | Programming.Guide,” accessed June 12, 2024, [https://programming.guide/hash-tables-open-vs-closed-addressing.html](https://programming.guide/hash-tables-open-vs-closed-addressing.html).
+* Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
--- a/notes/logic/equiv-trans.md
+++ b/notes/logic/equiv-trans.md
@ -621,8 +621,8 @@ END%%

 %%ANKI
 Basic
-Execution of `b[i] := e` of array $b$ yields what new value of $b$?
-Back: $b = (b; i{:}e)$
+Execution of `b[i] := e` of array $b$ in state $s$ yields what new value of $b$?
+Back: $b = (b; i{:}s(e))$
 Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
 <!--ID: 1713793130031-->
 END%%
--- a/notes/set/images/relation-ordering-example.png
+++ b/notes/set/images/relation-ordering-example.png
--- a/notes/set/index.md
+++ b/notes/set/index.md
@ -255,6 +255,14 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1717372494462-->
 END%%

+%%ANKI
+Basic
+Let $A$ and $B$ be sets. Proving the following is equivalent to showing what class is a set? $$\exists C, \forall y, (y \in C \Leftrightarrow y = \{x\} \times B \text{ for some } x \in A)$$
+Back: $\{\{x\} \times B \mid x \in A\}$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718105051820-->
+END%%
+
 ## Empty Set Axiom

 There exists a set having no members: $$\exists B, \forall x, x \not\in B$$
--- a/notes/set/relations.md
+++ b/notes/set/relations.md
@ -95,53 +95,130 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
 <!--ID: 1717678753145-->
 END%%

-Given two sets $A$ and $B$, the **Cartesian product** $A \times B$ is defined as: $$A \times B = \{\langle x, y \rangle \mid x \in A \land y \in B\}$$
+A **relation** $R$ is a set of ordered pairs. The **domain** of $R$ ($\mathop{\text{dom}}{R}$), the **range** of $R$ ($\mathop{\text{ran}}{R}$), and the **field** of $R$ ($\mathop{\text{fld}}{R}$) is defined as:
+
+* $x \in \mathop{\text{dom}}{R} \Leftrightarrow \exists y, \langle x, y \rangle \in R$
+* $x \in \mathop{\text{ran}}{R} \Leftrightarrow \exists t, \langle t, x \rangle \in R$
+* $\mathop{\text{fld}}{R} = \mathop{\text{dom}}{R} \cup \mathop{\text{ran}}{R}$

 %%ANKI
 Basic
-How is the Cartesian product of $A$ and $B$ denoted?
-Back: $A \times B$
+What is a relation?
+Back: A set of ordered pairs.
 Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
-<!--ID: 1717679397781-->
+<!--ID: 1718107987764-->
 END%%

 %%ANKI
 Basic
-Using ordered pairs, how is $A \times B$ defined?
-Back: $\{\langle x, y \rangle \mid x \in A \land y \in B\}$
+Are relations or sets the more general concept?
+Back: Sets.
 Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
-<!--ID: 1717679397797-->
+<!--ID: 1718107987776-->
 END%%

 %%ANKI
 Basic
-Who is attributed the representation of points in a plane?
-Back: René Descartes.
+How is the ordering relation $<$ on $\{2, 3, 5\}$ defined?
+Back: As set $\{\langle 2, 3\rangle, \langle 2, 5 \rangle, \langle 3, 5 \rangle\}$.
 Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
-<!--ID: 1717679397825-->
+<!--ID: 1718107987783-->
 END%%

 %%ANKI
 Basic
-Why is the Cartesian product named the way it is?
-Back: It is named after René Descartes.
-Reference: “Cartesian Product,” in _Wikipedia_, April 17, 2024, [https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305](https://en.wikipedia.org/w/index.php?title=Cartesian_product&oldid=1219343305).
-<!--ID: 1717679397836-->
+How is the ordering relation $<$ on $\{2, 3, 5\}$ visualized?
+Back:
+![[relation-ordering-example.png]]
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718107987794-->
 END%%

 %%ANKI
 Basic
-Suppose $x, y \in A$. What set is $\langle x, y \rangle$ in?
-Back: $\mathscr{P}\mathscr{P}A$
+A relation is a set of ordered pairs with what additional restriction?
+Back: N/A.
 Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
-<!--ID: 1717679397848-->
+<!--ID: 1718107987803-->
 END%%

 %%ANKI
 Cloze
-{$x \in A$} iff {$\{x\} \subseteq A$} iff {$\{x\} \in \mathscr{P}A$}.
+For relation $R$, {$xRy$} is alternative notation for {$\langle x, y \rangle \in R$}.
 Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
-<!--ID: 1717679397860-->
+<!--ID: 1718107987813-->
+END%%
+
+%%ANKI
+Basic
+How is ordering relation $<$ on set $\mathbb{R}$ defined using set-builder notation?
+Back: As $\{\langle x, y\rangle \in \mathbb{R} \times \mathbb{R} \mid x \text{ is less than } y\}$.
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718107987822-->
+END%%
+
+%%ANKI
+Basic
+How is $x < y$ rewritten to emphasize that $<$ is a relation?
+Back: $\langle x, y \rangle \in \;<$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718107987831-->
+END%%
+
+%%ANKI
+Basic
+How is the identity relation on $\omega$ defined using set-builder notation?
+Back: $\{\langle n, n \rangle \mid n \in \omega\}$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718107987840-->
+END%%
+
+%%ANKI
+Basic
+How is the domain of relation $R$ denoted?
+Back: $\mathop{\text{dom}}{R}$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718107987850-->
+END%%
+
+%%ANKI
+Basic
+How is the domain of relation $R$ defined?
+Back: $x \in \mathop{\text{dom}}{R} \Leftrightarrow \exists y, \langle x, y \rangle \in R$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718107987862-->
+END%%
+
+%%ANKI
+Basic
+How is the range of relation $R$ denoted?
+Back: $\mathop{\text{ran}}{R}$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718107987872-->
+END%%
+
+%%ANKI
+Basic
+How is the range of relation $R$ defined?
+Back: $x \in \mathop{\text{ran}}{R} \Leftrightarrow \exists t, \langle t, x \rangle \in R$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718107987880-->
+END%%
+
+%%ANKI
+Basic
+How is the field of relation $R$ denoted?
+Back: $\mathop{\text{fld}}{R}$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718107987887-->
+END%%
+
+%%ANKI
+Basic
+How is the field of relation $R$ defined?
+Back: $\mathop{\text{fld}}{R} = \mathop{\text{dom}}{R} \cup \mathop{\text{ran}}{R}$
+Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
+<!--ID: 1718107987897-->
 END%%

 ## Bibliography